**Author(s): ** Kristopher Williams**Journal: ** Mathematics ISSN 2227-7390

**Volume: ** 1;

**Issue: ** 1;

**Start page: ** 31;

**Date: ** 2013;

Original page**Keywords: ** line arrangement |

hyperplane arrangement |

Oka and Sakamoto |

direct product of groups |

fundamental groups |

algebraic curves**ABSTRACT**

Let C1 and C2 be algebraic plane curves in C2 such that the curves intersect in d1 · d2 points where d1, d2 are the degrees of the curves respectively. Oka and Sakamoto proved that π1(C2 C1 U C2)) ≅ π1 (C2 C1) × π1 (C2 C2) [1]. In this paper we prove the converse of Oka and Sakamoto’s result for line arrangements. Let A1 and A2 be non-empty arrangements of lines in C2 such that π1 (M(A1 U A2)) ≅ π1 (M(A1)) × π1 (M(A2)) Then, the intersection of A1 and A2 consists of /A1/ · /A2/ points of multiplicity two.

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